Page 45 - Intro to Space Sciences Spacecraft Applications

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Introduction to Space Sciences and Spacecraft Applications
The most important point this law reveals is that the speed of an object
in an orbit changes with changing distance between the bodies. For the
areas shown in Figure 2-3 to be the same, the orbiting body must slow
down when it is farther away so that the line between the bodies sweeps
out the same area in the same amount of time as when it is traveling clos-
er (both At’s shown are equal). When Newton’s laws are discussed, we
will find a useful expression which shows this dependency of velocity on
orbital distance.
Kepler’s Third Law. In his third law, Kepler reveals that a relationship
exists between the semi-major axis and the period of an orbit. Stated
mathematically:

In equation 2-6, T represents the orbital period, or time it takes to travel
through one complete orbit with semi-major axis a. The term p represents a
gravitational parameter which has a specific value for each body around
which an orbit may be described. The gravitational parameter for the earth is:

CI English units Metric units
Earth 1.4077 x 10l6 ft3/sec2 3.986 x lo5 km3/sec2

Use of this form of the gravitational parameter in Kepler’s time equa-
tion (eq. 2-6) with the proper units used for the semi-major axis results in
an orbital period given in seconds.

Example Problem:

With the knowledge that it takes the moon 27.32 (sidereal) days
to complete one orbit around the earth, determine the semi-major
axis of the moon’s orbit.
Note: a sidereal day is the time it takes for the earth to complete
one full rotation about its axis with respect to the (inertially fixed)
stars. A solar day, measured with respect to the sun, differs due to
the fact that the earth is also in motion around the sun.

1 sidereal day = 23 hrs, 56 min, 4 sec = 86,164 sec